Question

What is the power dissipated in an $LCR \, $ series circuit connected to an ac source of emf $\epsilon \, $ ?

• A. $\frac{\left(\epsilon \right)^{2} \sqrt{R^{2} + \left(\omega L - \frac{1}{\omega C}\right)^{2}}}{R}$
• B. $\frac{\left(\epsilon \right)^{2} \left[R^{2} \left(\omega L - \frac{1}{\omega C}\right)^{2}\right]}{R}$
• C. $\frac{\left(\epsilon \right)^{2} R}{\sqrt{R^{2} + \left(\omega L - \frac{1}{\omega C}\right)^{2}}}$
• D. $\frac{\left(\epsilon \right)^{2} R}{\left[R^{2} + \left(\omega L - \frac{1}{\omega C}\right)^{2}\right]}$

Answer 1

Answer: (D)

Average power, $P=V_{r m s}I_{r m s}cos \phi$
Here, $Z=\sqrt{R^{2} + \left(X_{L} - X_{C}\right)^{2}}cos \phi=\frac{R}{Z}$
But $I_{r m s}=\frac{V_{r m s}}{Z}. \, \therefore P=V_{r m s}^{2}\frac{R}{Z^{2}}$
$\therefore P=V_{r m s}^{2}\frac{R}{\left\{\right. R^{2} + \left(X_{L} - X_{C}\right)^{2} \left.\right\}}=\frac{\left(\epsilon \right)^{2} R}{\left[R^{2} + \left(\omega L - \frac{1}{\omega C}\right)^{2}\right]}$